рус eng
«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2016. Vol. 16

Bernoulli Polynomials in Several Variables and Summation of Monomials over Lattice Points of a Rational Parallelotope

Author(s)
O. A. Shishkina
Abstract

The Bernoulli polynomials for natural x were first considered by J.Bernoulli (1713) in connection with the problem of summation of the powers of consecutive positive integers. For arbitrary x these polynomials were studied by L.Euler. The term”Bernoulli polynomials” was introduced by Raabe (J.L. Raabe, 1851). The Bernoulli numbers and polynomials are well studied, and are widely used in various fields of theoretical and applied mathematics.

The article is devoted to some generalizations of the Bernoulli numbers and polynomials to the case of several variables. The concept of Bernoulli numbers associated to a rational cone generated by vectors with integer coordinates is defined. Using the Bernoulli numbers, we introduce the Bernoulli polynomials of several variables. Next we construct a difference operator acting on functions defined in a rational cone, and by methods of the theory of generating functions we prove a multidimensional analogue of the main property, which is the fact that the Bernoulli polynomials satisfy a difference equation.
Also, we calculate the values of the integrals of the Bernoulli polynomials over shifts of the fundamental parallelotope, and for the sum of monomials over integer points of a rational parallelotope we find a multidimensional analogue of the Bernoulli formula, where the sum above is expressed in terms of the integral of the Bernoulli polynomial over a parallelotope with variable ”top” vertex.
Keywords
Bernoulli numbers and polynomials, generating functions, summation of functions, rational parallelotope
UDC
517.55+517.962.2

MSC

32A05+11B68

References

1. Gelfond A.O. Calculus of finite differences (in Russian). Moscow, Nauka, 1977..2. Cassels J.W.S An introduction to the geometry of numbers (in Russian). Moscow, Mir, 1965.

3. Lando S.K. Lectures on generating functions(in Russian). Moscow, MZNMO, 2007.

4. Leinartas E.K., Rogozina M.S. Solvability of the Cauchy problem for a polynomial difference operator and monomial bases for the quotients of a polynomial ring (in Russian). Siberian Mathematical Journal, 2015, vol. 56, no 1, pp. 111-121.

5. Nekrasova T.I. Sufficient conditions of algebraicity of generating functions of the solutions of multidimensional difference equations (in Russian). Izvestiya Irkutskogo Gosudarstvennogo Universiteta. Seriya "Matematika", 2013, vol. 6, no 3, pp. 88-96.

6. Stanley R. Enumerative combinatorics (in Russian). Moscow, Mir, 1990.

7. Ustinov A.V A discrete analoge of Euler’s summation formula (in Russian). Mat. Zametki, 2002, vol. 71, issue 6, pp. 931-936.

8. Shishkina O.A. The Euler-Maclaurin formula for rational parallelotope(in Russian). Izvestiya Irkutskogo Gosudarstvennogo Universiteta. Seriya "Matematika", 2015, vol. 13, pp. 56-71.

9. Euler L. Differential calculus (in Russian), translation from lat., M.-L., 1949

10. Brion M., Berline N. Local Euler-Maclaurin formula for polytopes. Moscow Mathematical Society Journal, 2007, vol. 7, pp. 355-383.

11. Brion M., Vergne M. Lattice points in simple polytopes. Journal of the American Mathematical Society, 1997, vol. 10, no 2, pp. 371-392.

12. Brion M., Vergne M. Residue formulae, vector partition functions and lattice points in rational polytopes. Journal of the American Mathematical Society, 1997, vol. 10, no 4, pp. 797-833.

13. Leinartas E.K., Nekrasova T.I. Constant coefficient linear difference equations on the rational cones of the integer lattice. Siberian Mathematical Journal, 2016, vol. 57, no 1, pp. 74-85.

14. Shishkina O.A. The Euler-Maclaurin Formula and Differential Operators of Infinite Order. Journal of Siberian Federal University, Mathematics & Physics,;2015, vol. 8, no 1, pp. 86-93.


Full text (russian)